physicsadmathstuto.com
physicsadmathstuto.com Jauay 2009 2 a 7. Give that X = 1 1, whee a is a costat, ad a 2, blak (a) fid X 1 i tems of a. Give that X + X 1 = I, whee I is the 2 2 idetity matix, (b) fid the value of a. 14 *N34694A01428*
physicsadmathstuto.com Jauay 2009 10. A = 3 2 0 0 3 2, B = 0 1 1 0, C = 1 2 1 2 1 2 1 2 blak (a) Descibe fully the tasfomatios descibed by each of the matices A, B ad C. (4) It is give that the matix D = CA, ad that the matix E = DB. (b) Fid D. (c) Show that E = 3 3 3 3. (1) The tiagle ORS has vetices at the poits with coodiates (0, 0), ( 15, 15) ad (4, 21). This tiagle is tasfomed oto the tiagle OR S by the tasfomatio descibed by E. (d) Fid the coodiates of the vetices of tiagle OR S. (4) (e) Fid the aea of tiagle OR S ad deduce the aea of tiagle ORS. 24 *N34694A02428*
physicsadmathstuto.com a 2 5. R =, whee a ad b ae costats ad a > 0. a b (a) Fid R 2 i tems of a ad b. Give that R 2 epesets a elagemet with cete (0, 0) ad scale facto 15, Jue 2009 blak (b) fid the value of a ad the value of b. (5) 14 *M35146A01424*
physicsadmathstuto.com a 2 7. A =, whee a is a costat. 1 4 (a) Fid the value of a fo which the matix A is sigula. 3 2 B = 1 4 (b) Fid B 1. The tasfomatio epeseted by B maps the poit P oto the poit Q. Give that Q has coodiates (k 6, 3k + 12), whee k is a costat, Jue 2009 blak (c) show that P lies o the lie with equatio y = x + 3. 20 *M35146A02024*
physicsadmathstuto.com Jauay 2010 a 5 5. A =, whee a is eal. 2 a + 4 blak (a) Fid det A i tems of a. (b) Show that the matix A is o-sigula fo all values of a. Give that a = 0, (c) fid A 1. 10 *N35143A01024*
physicsadmathstuto.com Jauay 2010 1 1 9. M = 2 2 1 1 2 2 blak (a) Descibe fully the geometical tasfomatio epeseted by the matix M. The tasfomatio epeseted by M maps the poit A with coodiates (p, q) oto the poit B with coodiates (3 2, 4 2). (b) Fid the value of p ad the value of q. (c) Fid, i its simplest sud fom, the legth OA, whee O is the oigi. (d) Fid M 2. (4) The poit B is mapped oto the poit C by the tasfomatio epeseted by M 2. (e) Fid the coodiates of C. 22 *N35143A02224*
physicsadmathstuto.com Jue 2010 2a 3 2. M =, whee a is a eal costat. 6 a blak (a) Give that a 2, fid M 1. (b) Fid the values of a fo which M is sigula. 4 *N35387A0428*
physicsadmathstuto.com Jue 2010 6. Wite dow the 2 2 matix that epesets blak (a) a elagemet with cete (0, 0) ad scale facto 8, (b) a eflectio i the x-axis. (1) (1) Hece, o othewise, (c) fid the matix T that epesets a elagemet with cete (0, 0) ad scale facto 8, followed by a eflectio i the x-axis. 6 1 A = ad 4 2 (d) Fid AB. k 1 B =, whee k ad c ae costats. c 6 Give that AB epesets the same tasfomatio as T, (e) fid the value of k ad the value of c. 14 *N35387A01428*
physicsadmathstuto.com Jauay 2011 2. A = B 2 0 = 3 1, 5 3 5 2 blak (a) Fid AB. Give that C = 1 0 0 1 (b) descibe fully the geometical tasfomatio epeseted by C, (c) wite dow C 100. (1) Q2 (Total 6 maks) *N35406A0332* 3 Tu ove
physicsadmathstuto.com Jauay 2011 8. 2 2 A = 1 3 blak (a) Fid det A. (b) Fid A 1. (1) The tiagle R is tasfomed to the tiagle S by the matix A. Give that the aea of tiagle S is 72 squae uits, (c) fid the aea of tiagle R. The tiagle S has vetices at the poits (0, 4), (8, 16) ad (12, 4). (d) Fid the coodiates of the vetices of R. (4) 22 *N35406A02232*
physicsadmathstuto.com Jue 2011 3. (a) Give that 1 2 A = 2 1 blak (i) fid A 2, (ii) descibe fully the geometical tasfomatio epeseted by A 2. (4) (b) Give that B = 0 1 1 0 descibe fully the geometical tasfomatio epeseted by B. (c) Give that k + 1 12 C = k 9 whee k is a costat, fid the value of k fo which the matix C is sigula. 8 *P38168A0832*
physicsadmathstuto.com Jue 2011 5. A = 4 a whee a ad b ae costats. b 2, blak Give that the matix A maps the poit with coodiates (4, 6) oto the poit with coodiates (2, 8), (a) fid the value of a ad the value of b. (4) A quadilateal R has aea 30 squae uits. It is tasfomed ito aothe quadilateal S by the matix A. Usig you values of a ad b, (b) fid the aea of quadilateal S. (4) 14 *P38168A01432*
physicsadmathstuto.com Jauay 2012 blak 4. A ight agled tiagle T has vetices A (, 11), B ( 21, ) ad C ( 24, ). Whe T is tasfomed by the matix P = 0 1 1 0, the image is T. (a) Fid the coodiates of the vetices of T. (b) Descibe fully the tasfomatio epeseted by P. The matices Q = 4 3 2 1 ad R = 1 2 3 4 epeset two tasfomatios. Whe T is tasfomed by the matix QR, the image is T. (c) Fid QR. (d) Fid the detemiat of QR. (e) Usig you aswe to pat (d), fid the aea of T. 8 *P40086A0824*
physicsadmathstuto.com Jauay 2012 8. A = 0 1 2 3 blak (a) Show that A is o-sigula. (b) Fid B such that BA 2 = A. (4) 20 *P40086A02024*
physicsadmathstuto.com Jue 2012 2. (a) Give that blak A = 3 1 3 B ad = 4 5 5 1 1 1 2 0 1 fid AB. (b) Give that ad C = 3 2 D = 8 6, 5 2k 4 k, E = C + D whee k is a costat fid the value of k fo which E has o ivese. (4) 4 *P40688A0432*
physicsadmathstuto.com Jue 2012 9. M = 3 4 2 5 blak (a) Fid det M. (1) The tasfomatio epeseted by M maps the poit (2a 7, a 1), whee a is a costat, oto the poit (25, 14). (b) Fid the value of a. The poit has coodiates (6, 0). Give that O is the oigi, (c) fid the aea of tiagle ORS. Tiagle ORS is mapped oto tiagle by the tasfomatio epeseted by M. (d) Fid the aea of tiagle OR'S'. Give that A = 0 1 1 0 (e) descibe fully the sigle geometical tasfomatio epeseted by A. The tasfomatio epeseted by A followed by the tasfomatio epeseted by B is equivalet to the tasfomatio epeseted by M. (f) Fid B. (4) 24 *P40688A02432*
physicsadmathstuto.com Jauay 2013 4. The tasfomatio U, epeseted by the 2 2 matix P, is a otatio though 90 aticlockwise about the oigi. blak (a) Wite dow the matix P. (1) The tasfomatio V, epeseted by the 2 2 matix Q, is a eflectio i the lie y = x. (b) Wite dow the matix Q. (1) Give that U followed by V is tasfomatio T, which is epeseted by the matix R, (c) expess R i tems of P ad Q, (d) fid the matix R, (1) (e) give a full geometical desciptio of T as a sigle tasfomatio. 8 *P41485A0828*
physicsadmathstuto.com Jauay 2013 6. X = 1 a 3 2, whee a is a costat. blak (a) Fid the value of a fo which the matix X is sigula. 1 1 Y = 3 2 (b) Fid Y 1. The tasfomatio epeseted by Y maps the poit A oto the poit B. Give that B has coodiates (1, 7 2), whee is a costat, (c) fid, i tems of, the coodiates of poit A. (4) 14 *P41485A01428*
physicsadmathstuto.com Jue 2013 1. blak M = x x 2 3x 6 4x 11 Give that the matix M is sigula, fid the possible values of x. (4) 2 *P43138A0232*
A 1 = 1 2 (A 7I) physicsadmathstuto.com Jue 2013 8. blak A = 6 2 4 1 ad I is the 2 2 idetity matix. (a) Pove that (b) Hece show that A 2 = 7A + 2I The tasfomatio epeseted by A maps the poit P oto the poit Q. Give that Q has coodiates (2k + 8, 2k 5), whee k is a costat, (c) fid, i tems of k, the coodiates of P. (4) 24 *P43138A02432*
physicsadmathstuto.com Jue 2013 R
physicsadmathstuto.com Jue 2013 R
Futhe Pue Mathematics FP1 Cadidates sittig FP1 may also equie those fomulae listed ude Coe Mathematics C1 ad C2. Summatios = 1 = 1 2 3 = = 1 6 1 4 ( + 1)(2 + 1) 2 ( +1) 2 Numeical solutio of equatios The Newto-Raphso iteatio fo solvig f( x ) = 0 : x + 1 f( x ) = x f ( x ) Coics Paabola Rectagula Hypebola Stadad Fom y 2 = 4ax xy = c 2 Paametic Fom (at 2, 2at) ct, c t Foci (a, 0) Not equied Diectices x = a Not equied Matix tasfomatios Aticlockwise otatio though θ about O: cosθ siθ siθ cosθ Reflectio i the lie cos 2θ si 2θ y = (taθ ) x : si 2θ cos 2θ I FP1, θ will be a multiple of 45. 8 Edexcel AS/A level Mathematics Fomulae List: Futhe Pue Mathematics FP1 Issue 1 Septembe 2009
Coe Mathematics C1 Mesuatio Suface aea of sphee = 4π 2 Aea of cuved suface of coe = π slat height Aithmetic seies u = a + ( 1)d S = 2 1 (a + l) = 2 1 [2a + ( 1)d] 4 Edexcel AS/A level Mathematics Fomulae List: Coe Mathematics C1 Issue 1 Septembe 2009
Edexcel AS/A level Mathematics Fomulae List: Coe Mathematics C2 Issue 1 Septembe 2009 5 Coe Mathematics C2 Cadidates sittig C2 may also equie those fomulae listed ude Coe Mathematics C1. Cosie ule a 2 = b 2 + c 2 2bc cos A Biomial seies 2 1 ) ( 2 2 1 b b a b a b a a b a + + + + + + = + K K ( N) whee )!!(! C = = < + + + + + + = + x x x x x 1, ( 2 1 1) ( 1) ( 2 1 1) ( 1 ) (1 2 K K K K R) Logaithms ad expoetials a x x b b a log log log = Geometic seies u = a 1 S = a 1 ) (1 S = a 1 fo < 1 Numeical itegatio The tapezium ule: b a x y d 21 h{(y 0 + y ) + 2(y 1 + y 2 +... + y 1 )}, whee a b h =